Superfluid Density of Neutrons in the Inner Crust of Neutron Stars: New Life for Pulsar Glitch Models

Calculations of the effects of band structure on the neutron superfluid density in the crust of neutron stars made under the assumption that the effects of pairing are small [N. Chamel, Phys. Rev. C 85, 035801 (2012)] lead to moments of inertia of superfluid neutrons so small that the crust alone is insufficient to account for the magnitude of neutron star glitches. Inspired by earlier work on ultracold atomic gases in an optical lattice, we investigate fermions with attractive interactions in a periodic lattice in the mean-field approximation. The effects of band structure are suppressed when the pairing gap is of order or greater than the strength of the lattice potential. By applying the results to the inner crust of neutron stars, we conclude that the reduction of the neutron superfluid density is considerably less than previously estimated and, consequently, it is premature to rule out models of glitches based on neutron superfluidity in the crust.

Figure 1

Superfluid number density in terms of the total number density as a function of $K/2k_F$, where $k_F$ is the Fermi wave number of the uniform noninteracting Fermi gas at the same density. The curves are for a lattice potential $V_K=0.25(K/2k_F{)}^2{E}_F$ with $E_F=k_F^2/2m$. The lines are for the case of $\mathrm{\Delta }\to 0$ (green dashed line), $\mathrm{\Delta }\approx 0.0464E_F$ (brown dashed-dotted line), $\mathrm{\Delta }\approx 0.208E_F$ (blue dotted line), and $\mathrm{\Delta }\approx 0.686E_F$ (red solid line).

Figure 2

Superfluid density $n_{zz}^s/n$ as a function of $\mathrm{\Delta }/|V_K|$ for the same lattice potential as in Fig. 1. The red solid line is for $K=2k_F$ and the blue dashed line is for $K=\sqrt{8/3}{k}_F\approx 1.633k_F$.

Figure 3

Lattice potential $V_K$ as a function of $K/2k_n$ from the work of Ref. [18]. Because there are so many reciprocal lattice vectors with $K<2k_n$, $K$ is treated as a continuous variable.